TIFR, Mumbai
March 9, 2017
Generalized Casimir Operators: Let $\mathfrak g$ be symmetrizable Kac-Moody Lie algebra and let $\Omega$ be Casimir operator. Let $A$ be commutative associative finitely generated algebra with unit. In this talk we consider highest weight modules for $\mathfrak g \otimes A$. We define operators $\Omega (a,b)$, $a, b \in A (\Omega (1,1)=\Omega)$ which operate on $\mathfrak g \otimes A$ module and commutes with $\mathfrak g$ action. We will specialize on tensor product modules (for $\mathfrak g$) which can be given a $\mathfrak g \otimes \mathbb C[t,t^{-1}]$ module structure.
We will use similar ideas for $gl_{N} \otimes A$ and construct more operators which commute with $gl_N$. It is well known that the center of $U(gl_N)$ is finitely generated as an algebra. Around 1950 Gelfand defined central elements (called Gelfand invariants) $T_k$ for each positive integer $k$. It is known that first $N$ generates the center. The decomposition of tensor product module for reductive Lie algebra is a classical problem. It is known that each Gelfand invariant acts as scalar on an irreducible submodule of a tensor product module. In this talk we define, for each $k$, several operators on the tensor product which commute with $gl_N$ action and they are not scalars. This means these operators take one highest weight vector to a new highest weight vector. Thus it is a practical algorithm to produce more highest weight vectors once we know one highest weight vector. Further the sum of these operators is $T_k$.
If time permits I will also talk about affine case.